Precalculus Examples

$g (x) = \ln (x)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 1.2

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \ln (1)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

The natural logarithm of $1$ is $0$ .

$f (1) = 0$

Step 2.2.2

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \ln (2)$

Step 3.2

The final answer is $\ln (2)$ .

$\ln (2)$

Step 3.3

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \ln (3)$

Step 4.2

The final answer is $\ln (3)$ .

$\ln (3)$

Step 4.3

Convert $\ln (3)$ to decimal.

$y = 1.09861228$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.69314718), (3, 1.09861228)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.693 \\ 3 & 1.099 \end{array}$

Step 6